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Summary: Isolated Summary: Isolated Systems, Temperature, Free Systems, Temperature, Free EnergyEnergy
Zhiyan WeiES 241: Advanced Elasticity5/20/2009
Isolated SystemsIsolated SystemsStatistical description of systems
Internal variable of an isolated system
The second law of thermodynamics
Entropy
Statistical DescriptionStatistical DescriptionSpecification of the state of the
system
Statistical ensemble
The fundamental postulate
Probability calculations
Statistical DescriptionStatistical DescriptionSpecification of the state of the
system Microscopic scale
Quantum description: a set of quantum numbers
Classical description: a phase point in the phase space
Macroscopic scaleA subset of quantum states of an isolated
system is called a macrostate (conformation, thermodyanamic state, or configuration)
Described by macroscopic parameters
Statistical DescriptionStatistical DescriptionSpecification of the state of the
system System
Any part of the world
Isolated systemA system is said to be isolated if it does not interact with the rest of the world– thermally isolated, mechanically isolated….
Statistical DescriptionStatistical DescriptionStatistical ensemble A very large number of identical
systems prepared under identical macroscopic conditions– same macroscopic state
Ergodic TheoremThe average behavior of a system over sufficient amount of time is the same as the average behavior of many identically prepared sytems.
Statistical DescriptionStatistical DescriptionThe fundamental postulate
An isolated system isolated for an enough long time is equally likely to be found in any of its quantum states!
Statistical DescriptionStatistical DescriptionProbability calculations The macrostates has ΩA number of
quantum states Ω is the number of quantum states of
an isolated system Probability for the isolated system to
be in macrostate A is
Statistical DescriptionStatistical DescriptionProbability calculations– examples Irreversible change in an isolated system–
half glass of wine. Evaporation is spontaneous, but not all the gas molecules will go back to the liquid again, why?
Dispersion of a drop of ink in a glass of wine
Ω ~VN
V– volume of the glass of wineN– number of ink particles
Internal Variable of An Internal Variable of An Isolated Isolated SystemSystemA function that maps a quantum
state of an isolated system to a number. That is, the domain of the function is the set of the quantum states of the isolated system, and the range of the function is a real number.
Example: half glass of wine!
Second Law of Second Law of ThermodynamicsThermodynamicsFor a thoroughly isolated system
that evolves from one macroscopic state to another, its entropy tend to increase!
TemperatureTemperatureThermal contactDefinition of absolute temperatureExperimental determination of
temperatureExperimental determination of the
number of quantum statesHeat capacity and latent heat
Thermal ContactThermal ContactOnly energy exchange between
two systems is allowedHeat transferEmpirical observations about
hotness:Two system will reach thermal equilibrium
in thermal contact after a long timeZeroth law of thermodynamicsLevels of hotness are ordered Levels of hotness are continuous
Definition of Absolute Definition of Absolute TemperatureTemperature
What is the most probable partition of energy?
A’ Ω’(U’) A’’ Ω’’(U’’)
Energy
dU
Isolated system
Definition of Absolute Definition of Absolute TemperatureTemperatureBefore energy exchange, the total
number of quantum states:
After the energy of the composite is partitioned as U’+dU and U’’-dU, # of quantum states:
The #s of states differ by
Experimental Determination Experimental Determination of Temperatureof Temperature
Calculate the temperature of a simple system by counting the number of states
Use the simple system to calibrate a thermometer by thermal contact
Use the thermometer to measure temperatures of any other system by thermal contact.
Experimental Determination of Experimental Determination of TemperatureTemperatureIdeal gas
T
P
V
VU
),(log
VNUNfNVU
VUNfNVU N
log),(log),,(log
),(),,(
Experimental Determination Experimental Determination of The Number of Quantum of The Number of Quantum StatesStates
Determine the function Ω(U) of a system up to a multiplicative factor. To fix the multiplication factor, we set Ω=1 as T 0, which is the Third Law of Thermodynamics.
Heat Capacity and Latent Heat Capacity and Latent HeatHeatHeat Capacity
Latent Heat
VVV T
ST
T
UC
PP
P T
ST
T
HC
Free EnergyFree EnergyA system with variable energy
A system with variable energy and an internal variable
Free energy
Co-existent phases of a substance
A System with Variable A System with Variable EnergyEnergyOpen a system: the system can
vary its energy U by thermal contact with the rest of the world
When the energy U is fixed at a particular value, the system becomes isolated
Characterized by Ω(U), S(U) and T(U)
A System with Variable A System with Variable EnergyEnergyLeading characteristics of the
curves
The horizontal position: no empirical significance
The vertical position: constricted by the 3rd Law of thermodyanics
A System with Variable A System with Variable EnergyEnergyThe function S(U) is usually
convex Two identical systems, each with
energy U Each part can exchange energy. U-Q
and U+Q
A System with Variable A System with Variable Energy and An Internal Energy and An Internal VariableVariableEntropy S(U,Y)
At a constant U, the most probable Y maximizes S(U,Y)
Free EnergyFree Energy (U,Y) specifies a macrostate of the composite
The entropy of the macrostate of the composite is
The above maximization is equivalent to the minimization below
Free EnergyFree Energy Temperature and entropy is one to one function Helmholtz free energy
An alternative way to introduce the free energy
The free energy of the system is the total energy of the composite of the system and the thermostate in thermal equilibrium
Co-existent phases of a Co-existent phases of a substancesubstanceN– number of molecules in one phaseThe entropy per molecule is The energy per molecule is Two phases
Co-existent phases of a Co-existent phases of a substancesubstanceExamine co-
existent phases using the function u(s)
Examine co-existent phases using the free energy
Phase Transition of The Phase Transition of The Second Kind Second Kind A crystal has a rectangular symmetry at high
temperature